[6, 5919] finitely based
[6, 5919] is a semigroup with 3 idempotents and a zero, 1. It is finitely based: 3 identities define its variety, which no other semigroup of order six generates.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 | 1 | 1 |
| 3 | 1 | 1 | 1 | 1 | 1 | 1 |
| 4 | 1 | 1 | 1 | 4 | 4 | 6 |
| 5 | 1 | 1 | 2 | 4 | 4 | 6 |
| 6 | 1 | 1 | 1 | 4 | 4 | 6 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal; grey: the zero.
Structure
- Smallsemi
- SmallSemigroup(6, 5919)
- Idempotents
- 1, 4, 6
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 5
- Rank
- 3, generated by {3, 5, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 3 identities irredundant
- x² ≈ x³
- xyz ≈ yxz
- xyz ≈ xyz²
Irredundant: none of these identities follows from the others; for each, a semigroup satisfies the others but not it. Obtained from a basis certified in Lean by removing identities that follow from the others, with the prover Vampire; the equivalence rests on Vampire's proofs, not on Lean. Shortest known, not known to be minimal.
Basis certified in Lean: 5 identities
- x² ≈ x³
- x²y ≈ xy²
- xyx ≈ yx²
- xyz ≈ yxz
- xyz ≈ xyz²
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6DirectPowerAvailableV3.S6_6155_Aeb958c93e553_Part01.S6_5919.representative_basis
BasisFor Generated.Order6DirectPowerAvailableV3.S6_6155_Aeb958c93e553_Part01.S6_5919.table.semigroup Generated.Order6DirectPowerAvailableV3Sources.S6_6155.sourceBasis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- direct-power transfer. A semigroup S with a certified basis lies in the variety of this one, as a subsemigroup of a direct power of this one, and this semigroup satisfies the basis of S. Then every identity of this semigroup holds in S and follows from the basis, so the basis of S is a basis here too.
- Size
- Checking this class alone compiles 27 Lean files with 12,295 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 145 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_5919 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_5919 is the table, with the elements numbered 0 to 5.
Variety
[6, 5919] generates the variety V[5919]; no other semigroup of order six generates it, and 435 semigroups of order six lie in it. [6, 5919] lies in 80 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[5919] in the inclusion graph, with the varieties above and below it.
Nearby tables
Semigroups whose Smallsemi table differs from this one in one or two entries (row, column), as the tables are listed, not up to renumbering.